00336138dv

IEEE Transactions on Power Systems, Vol. 9, No. 3, August 1994 1285

DETECTION OF TRANSIENTLY CHAOTIC SWINGS IN POWER SYSTEMS USING REAL-TIME PHASOR MEASUREMENTS

Chih-Wen Liu James S. Thorp Jin Lu Robert J . Thomas, Hsiao-Dong Chiang Cornell University ORA Coporation Cornell University Ithaca, N.Y. 14853 Ithaca, NY 14850 Ithaca, N.Y. 14853

Abstract In this paper, the concepts of transiently chaotic swings and windowed Lyapunov exponents in power system dynamics are described. An efficient computer method to detect a transiently chaotic swing from a set of real-time phasor measurements is presented. The method estimates the largest windowed Lyapunov exponent, A[NATl as a detection index. In simulation results, it is shown that the proposed method has the potential to detect a transiently chaotic swing and the estimated X/NA?l can be used as an on-line stability index to predict multi-swing transient instability.

1 Introduction Electric power systems in this country are undergoing sig- nificant changes. Due to ever-incresasing environmental and economical pressures, it becomes extremely difficult to con- struct new transmission or generation facilities. As a result of the limited power supply and steady growth of load de- mand, power systems are increasing susceptible to abnormal conditions following a severe disturbance. Therefore, the operating point is pushed closer to the stability boundary and the issue of maneuvering a system to a more secure operating condition by on-line stability monitoring and control becomes a critical task. With t,he advent of recently advanced phasor measurement techniques[3], there are possibilities for developing new approaches to assessing power system dynamics in a real-time enviroment[l][2][13]. For example, real- time phasor measurements promise to be used in detection of transient instability for out-of-step relaying or control[l][4]. More specifically, use can be made of these measurements to determine whether a given in-progress transient swing is one from which the system will or will not recover and promptly initiate necessary relaying function such as out- of-step blocking and triping, load shedding and restoration, dynamic braking and modulation of HVDC link power flow. For the purpose of achieving a more secure real-time operation of a power system under stressed condition, we need to develop methods that can assess on-line opearting conditions and produce necessary information for controls in time.

The discovery of complicated dynamical behavior(chaos) in the two-generator system[8][5] and the randomness of tra- jetories near stability boundary in the Florida-Georgia two- machine equivalent system[7] together with the availability of real-time phasor angle measurement motivate our investi- gation of on-line detection techniques of transiently chaotic swings(transient chaos) in power system dynamics. Our fo- cus will be on developing computation algorithm to detect transient chaos following a severe disturbance. We first de- fine the concepts of transiently chaotic swings in section 2, and then briefly describe the techniques of phasor measurement in section 3. In section 4, we describe how to use the measurements to detect transiently chaotic swings. Finally, simulation results are shown to demonstrate the effectiveness of the proposed method.

2 Transiently Chaotic Swings In the context of nonlinear dynamics, chaos means long-term irregular or random but bounded trajectories in a deterministic dynamical system which are very sensitive to initial condition(S.1.C.). In other words, a trajectory which is chaotic is unpredictable, even though the trajectory envolves accord- ing to deterministic equations. Since the discovery(within the last three decades) of chaos, chaotic dynamics has become one of the most exciting topics in nonlinear system research and has far-reaching consequences in many branches of science, for example, fluids near the oneset of turbulence, laser, chemical reactions and classical many-body systems etc. So far, due to theoretical results and t+ availability of high-speed but inexpensive computer systems, it has become clear that chaos is abundant in nature.

For power system dynamics, the numerical discovery of chaos is a recent affair[8][5][6]. Since the major characteristic of chaos is unpredictability, chaos is definitely not an allowed pheonomena from a dynamic security point of view. In particular, when a power swing is chaotic following a large disturbance, it can be mistaken as a stable one by classical first swing criterion, but i t might randomly give rise to a loss of synchronism. On the other hand, the broad-band spectrum property of a chaotic power swing will cause harmonic prob- lems. In the case of transient stability problem, the former situation is preferred to the latter situation. In this paper, we consider on-line detection of transiently chaotic swings

93 SM 526-4 PWRS by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation

Canada, July 18-22, 1993. 30, 1992

A paper recommended and approved

at the IEEEIPES lgg3 Summer Meeting* Vancouver* B'c' ' which are conceptly different from the traditional definition Manuscript submitted Dec.

made available for printing Kay 3 , 1993. of chaos in the nonlinear texts. Now we give a definition of transient chaotic swings(transient chaos) as follows:

PRINTED IN USA

0885-8950/94/$04.00 0 1993 IEEE

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Definition of transient chaos: Given an observation window, a transient swing is defined as transient chaos if and only if the characteristic of the transient swing is the same as that of chaos over the entire window.

The above definition is conceptual. It can be made precise by the introduction of the Lyapunov exponents.

Lyapunov exponents can be used to measure the sensitive dependence on initial states for chaotic systems. Lyapunov exponents are measures of the rate of divergence (or conver- gence) of trajectories which are initially infinitesimally s e p arated. A mathematical definition of Lyapunov exponents can be made in terms of the variational equation of nonlinear dynamical system ,x = f(z). First we differentiate equation i. = f ( z ) with respect to initial point,zo to get

Dz, i t (zo) = Dlf(dt(zo))Dz, d t ( Z O ) , (1)

Dlodto(zO) = I , where $ t ( z ~ ) is the solution of 3 = f(z) from 20.

Defining D , , # J ~ ( ~ O ) = @t(zo) , rewrite Eqn.1 as

Eqn.2 is the so-called variational equation which is a linear time-varying equation describing the evolution of @t(zo).

Let mi(b), i = 1, 2, ..., n, be the eigenvalues of @t(zo). A precise definition of Lyapunov exponents is given, X i , i = 1 , 2 , . . . , n by the following equation:

From this definition, it is intuitive that the Lyapunov exponents represent the average rate of expansion or contraction of volume of the i-th dimension in R" phase space on the attractor. That is, if it is positive, then the difference between initial conditions will expand in a particular direction along the trajectory. Otherwise, the difference will contract in a particular direction along the trajectory. For a stable steady-state behavior, contraction must outweigh expansion; so cy=, Xi < 0.

The Lyapunov exponents of a chaotic trajectory have at least one positive X i which corresponds to the sensitive dependence feature. So, the largest Lyapunov exponent, X i , where we order Xi in a desending way, of a chaotic trajectory must be positive. This feature distinguishes chaos from the other types of steady-state behaviors. For the other types of steady-state behavior, equilibrium points and periodic solution, all Lyapunov exponents are negative. Let us illus- trate the possible types of Lyapunov exponents of chaos in three-dimensional and four-dimensional systems. In a three- dimensional case, the only possibility for the Lyapunov exponents is of type (+$,-), that is, XI > 0, & = O , and A3 < 0. In a four dimensional case, there are two possible cases: ( + , O , - , - ) and (+,+,O,-). The latter case is so-called hyperchaos. The possible types of Lyapunov exponents in higher dimensional systems can be derived in a similar way with the constraint

of Xi < 0. Using the Lyapunov exponents quantitity, we can redefne

transient chaos in a more precise way:

Given an observation window NAT, the windowed Lyapunov exponents, X i N A T ] , i = 1, . . . , n are defined as:

(4) 1

X ~ N P T I := ln(lmi(t)l),

where mi(t) is the same as that of Eqn.(B). A transient swing is said to be transient chaos over the window if and only if it' has at least one positive X i N A q .

Remark 1: A,easy consequence of the definition is that as NAT -+ 00, X i N A T ] + Xi. so, if a swing is chaotic, then it must be transiently chaotic. However,the reverse statement is not true in general. Remark 2: A transiently chaotic swing has the characteristics of broad-band power spectrum and intrinsic unpredictability over the window, since these properties are natu- ral consequences of having positive windowed Lyapunov exponents which account for the extreme sensitivity to initial conditions. Remark 3: One advantage of using windowed Lyapunov exponents is that in the case of transient stability it is useful to calculate windowed Lyapunov exponents for a small window to study the nature of short-term swing.

3 Real-Time Phasor Measure- ments

Microprocessor-based relays obtain lower level signals from the current and voltage transducers, sample the current and voltage signals, perform calculations on the samples, and reach relaying decisions. There are a number of algorithms that have been developed for protection of transmission lines, power transformers, and buses[9],[?]. For the purpose of the measurement systems of interest here, the Fourier-type algorithms are most important. That is, the Fourier-type algorithms are necessary to produce the desired measurements.Let y(t) represent a voltage or current in sine-cosine form where ~ ( t ) represents noiselike signals.

y ( t ) = Y, coswot + Y, sin wot + c ( t ) (5)

Estimates of the values Y, and Y, can be obtained with a Fourier calculation where there are N samples per cycle(or half-cycle) of the fundamental frequency wo(the nominal power system frequency, 50 or 60 Hz)[9].

N - I

Yc = d y(nAT) cos wonAT (6) n=O

= $ E y(nAT) sin wonAT (7) l l = O

If y ( t ) is a pure sinusoid, which equals cos(wot+6), the complex number Y , computed from Eqns.(6) and (7), has the

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, where t denotes transpose, k = 0,1,. , N , A T is the sain- pling period, N A T is the length of the measurement window, and m is the number of generators under study.

angle 6. If y ( t ) is a bus voltage then the resulting complex voltage phasor can be thought as the state of the system for many applications. As the window of N samples moves in time [the sums in Eqns.(t) and (7) taken from n = k to n = N + k - 11 the angle of Y rotates. A reference angle can be established and the calculations can be made recursively by writing the equation where 4 equals WO A T and Y L is the phasor computed using N sample values ending at sample L .

The recursive calculation is computationally efficient since only one multiplication is performed in Eqn.(8). Even more importantly, if the sign9 y(t) is a pure sinusoid at the nominal frequency W O , then Y L is stationary in phase with a phase angle equal to the angle 6 a t the instant a t which the recur- sion was begun. If the frequency is WO plus Aw (the power system is off-nominal frequency because of a generation-load imbalance), the computed phasor rotates at a rate Aw pro- viding a highly accurate frequency measurement device[?]. That is, the frequency can be measured by watching the computed phasor rotate. It is assumed that the sampling rate is fixed, i.e., A T equals E.

Currently one limiting factor to this technology is the availability of an accurate sampling clock synchronism system. The use of a navigation broadcast system such as the Global Positioning System(GPS) has made it possible to produce synchronizing pulses once every second with accuracy of 1 psec. With such accuracy it is possible to obtain accurate voltage phasor measurements during fast changes of system operation such as transient swings.

4 Detection of Transiently Chaotic Swings

From section 2, one knows that a swing is transiently chaotic if and only if there exist at least one positive windowed Lya- punov exponents. Let AtNaTl be the largest windowed Lya- punov exponent. Then it is enough and computational efficient to employ A/,,,] as the detection index, since it dom- inates the property of sensitivity to initial conditions. More specifically, if the sign of the calculated XiNaTl is positive, then the swing is identified as transiently chaotic and larger quantity of X~N,q implies more sensitivity to initial conditions. Based on the above concepts, a two-step computation algorithm is developed to detect transiently chaotic swings.: Corn put at ion Algor i t hrn

We assume that the power system is equipped with phasor measurement device at each generator and the transient post-fault dynamics is governed by the classical swing equations. The post-fault time variation of generator angles and frequencies with respect to center of angle are obtained from Eqn.8. These real-time phasor measurements are commu- nicated to control center to perform detection task. The algorithm is outlined as follows:

Given a set of angle measurement vector {6(kAT)} = ((Si(kAT), ..., Sm(kAT))}t and a set of frequency measurement vector {w(kAT)} = {(wl(kAT), ... ,wm(kAT)}t

Step 1 : Using {6(kAT)}and {o(kAT)} to identify the classical swing equations.

Step 2: Estimating A/,,q from the identified classical swing equations. Using the sign of the estimated XiNaTl as a detection index.

Explanation of Step 1:Assuming that the system consists of m generators, the classical swing equations are of the forms:

m 1 6;. = -(Pi - Diwi - EiEjBij sin(6, - ~ 5 ~ )

j=1 Mi m

- EiE,Gi, cos(& - 4) ) (9) 3=1

where 6, is the i-th component of a(,) and P,,D,,B,,, GI,, M,, E, are real power injections, damping, , susceptance, conductance, moment of inertial and constant voltage of generator 1 , respectively. BY setting P, = 9, D, = Mt '

are system parameters to be estimated by means of minimiz- ing the following sum of squared errors:

B,, E t E d, B, L , and G,, = E 1 ~ , G ' 3 , then P,,D,,B,, and G,,

N-1

where

m

ei(kAT) = ii(kAT) - Pi + Diwi + C B i j j=1

m

sin(& - 6,) + C G i j cos(6i - Sj) (11) j=1

and &(kAT) can be approximated by the consecutive frequency measurements:

The summation ranges from 0 to N - 1, because &(NAT) is not available in Eqn.12. The solution to this linear least- square error problem can be approached by solving normal equation derived from Eqn.lO.[l2] Explanation of Step 2: Let = j ( 6 ) be the identified swing equation from step 1. The corresponding variational equation can be found from Eqn.2. Randomly choose an initial perturbation As(0) near to initial measurement angle vector a(0). Set A$(') = A$(O),u(') = A$(O)/llA6(O)ll, and 6 ( O ) = 6(0), where 1 1 0 1 1 is Euclidean 2-norm. Integrate the

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variational equation from U(’) for A T seconds to obtain

where @AT(.) is the one-step, AT, transition matrix of the variational equations of the identified swing equations. Let dAT) = A6(AT)/llA8(AT)ll be the normalized version of Ad(AT). Integrate the variational equation from dAT) for A T seconds again to obtain

A8(2AT) = @AT(b(AT))uL(AT) where is the next sampling measurement. Repeat this integration/normalization procedure over the whole window N times. Then the estimated largest windowed Lyapunov exponent is computed as follows:

N

k=l

Practical Considerations 1.Reference F’rame:In the transient case, each rotor angle is compared with the center of angle(C0A) reference for transient chaos detection. In a real-time enviroment it is impossible t o acquire all angle measurements in a short period of time. On the other hand, our objective is not to determine the stability of the whole system, but to detect transient chaos for relay and controller decisions in related areas. Therefore, only nearby unit angles are required to compute a ”local COA” to decide if any action should be initiated in this part of the network. 2. Length of the W i n d o w NAT: Since we want to detect transiently chaotic swing as early as possible, the window should be chosen as short as possible from this point. On the other hand, we need enough measurements to identify the system and obtain more accurate A[,,,]. Therefore, there is a trade-off between these things. A reasonable choice is that the dimension of normal equations, which equals to N , is about twice than the number of unknown parameters. 3. Noise: Due to the inevitible presence of noise in measurements and numerical computation, it is safer to choose a threshhold value y to compare wit,h ifNAT1. That is, if ;\tNAT1 > y, then the swing is transiently chaotic or else it is not. Usually y can be chosen by hypothesis testing.

5 Simulation Results The proposed method was applied to a three machine system with lossless transmission lines. The reduced network rep- resentted by only connections with internal machine nodes is shown in Fig.1. The first case consists of two machines with high damping and the rest one is treated as slack or a machine with an almost infinte capacity to absorb or supply power to the system. The second case is a special three- machine system without damping and two machines have larger inertias than the rest one, but no machine is treated as slack. The simulated phasor measurements are obtained from direct integration of the classical swing equations. The

Figure 1: Three Machine System

measurements are assumed to be obtained at a rate of once every O.Olsec, although the real-time phasor measurements are performed constantly at a sampling rate, say, 12 times per cycle(1/720 sec)[9]. At a rate of 0.01 sec there is overlap in the raw data used for consecutive phasor estimates. The measurement window, NAT, is set to be 1 sec and and the threshhold value, y, is set to be zero. Case 1:

In this first case, machine 3 is treated as slack bus. The classical swing equations describing the post-fault system are as follows:

The system parameters are set to be Pi = 0.59739,& = l.O,& = 0 . 5 , P ~ = 0.89739,& = 1.0,& = 0.3 and BIZ = I321 = 0.1 By [5], the system has chaotic swings with the above specified parameters. The simulated time response of angles and angular frequencies of two machines are integrated from four initial conditions, deviated from equilibrium point, which model 4 different disturbances. Appling the proposed method to the simulated measurements, we obtain the estimated system parameters and XtlJecl, shown in table 1. From the table, we conclude the first two swings caused by disturbances 1 and 2 are transiently-chaotic swings over l(sec.) window by the positive sign of X~l,e,l. Fig.2-5 show the swing curves of angular frequency of machine 1 corresponding to four disturbances over 20 seconds from direct integrations. From Fig.2 and Fig.3, we can see the transiently chaotic time-behavior and sensitive dependence on the initial conditions of transient swings caused by disturbance 1 and 2. From Fig.4 and Fig.5, however, we can’t find the similar characteristics from transient swings caused by disturbance 3 and 4. It should be noted that the method is applied to a smaller measurement window, 1 sec, but the conclusions of the nature of the transient swings still hold for a larger simulation window, 20 sec. These results show the effectiveness of using the sign of AflSec1 as a detection index.

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(1.2,0.1,1.4,0.5) disturbance 4: initial condition= (0.2,0.1,0.5,0.8)

11 " " " ' " ' 0 2 4 6 8 10 12 14 16 IS 20

sec

..

Dl = 0.50 8 2 = 0.34 B2l = 0.09 "tsecl = -0.11 4 = 0.58 4 = 0.89 I 3 2 = 0.09 8 1 3 = 0.98 B23 = 0.99 D1 = 0.49 D 2 = 0.3 Bzl = 0.09

Figure 2: Transiently chaotic time-behavior and sensitive dependence on the initial conditions of transiently chaotic swing of angular frequency of machine 1 caused by disturbance 1. The solid curve and dotted curve are integrated from two close initial conditions:solid curve:(2.6 0.1 1.4 1.5),dotted curve:(2.8 0.1 1.4 1.5)

Table 1:Estimated System Parameters and

In this second case, machine 1 and machine 2 have larger inertials than machine 3, i.e., M 1 - - a, M2 = with A?fl,A?2 ,and M3 of order 1. c is a small positive number. Under the assumptions of conservative of energy and mo- mentum , the system can be described by the following reduced swing equations, for more details see [6]

& = w1

83 = w3

LA = A - E 1 2 s in( ( l+ p1)61 + ~ ~ 3 6 3 ) - 813 sin(& - 63)

j 3 = P3 - 831 sin(& - 61)

- 8 3 2 sin(pl61 + (I + cp3)53) (15)

In our simulation, A, P3, 8 1 2 , B13, &I and B32 are unknown system parameters to be estimated and 6 = 0.01, p i = a = p3 = = 0.707 are known parameters. This sys- MZ M2 tem was shown theoretically to exist chaotic motions using Melikov's theorem[6]. In our simulation, we obtained the simulated measurements by integrating Eqn.15 using the parameter values PI = 0.59739, P 3 = 0.89739, = 1.0, 813 = 0.001, 831 = 0.1, 8 3 2 = 1.0 and initial condition

I . . . . . . . ' ' J '0 2 4 6 8 10 12 14 16 18 20

105

Figure 3: Transiently chaotic time-behavior and sensitive dependence on the initial conditions of transiently chaotic swing of angular frequency of machine 1 caused by disturbance 2. The solid curve and dotted curve are integrated from two close initial conditions: solid curve:(2.8 0.1 0.5 0.8),dotted curve:(3.0 0.1 0.5 0.8)

DLturburo 3 ais

. . . . 0.1

0.3 '%, #'

-a35 . .

0 2 4 6 8 10 12 14 16 18 20

Ec

Figure 4: Nonchaotic time-behavior and correlative dependence on the initial condions of angular frequency of machine 1 caused by disturbance 3. The solid curve and dotted curve are integrated from two close initial conditions: solid curve:(1.2 0.1 1.4 0.5),dotted curve:(1.4 0.1 1.4 1.5)

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0 2 4 6 8 10 12 14 16 18 U)

Figure 5: Nonchaotic time-behavior and correlative dependence on the initial conditions of angular frequency of machine 1 caused by disturbance 4. The solid curve and dotted curve are integrated from two close initial conditions: solid curve:(0.2 0.1 0.5 0.8),dotted curve:(0.4 0.1 0.5 0.8)

(61(0),w1(0), 63(0),w3(0) = (0 .3,0.0,0.3,0.0) . The method was used on these simulated measuremeyts. The estimated parameters and i/lsec] are PI = 0.5878, P3 = 0.8988, B12 = 1.0051,813 = 0.1507, &1 = 0.0957,&2 = 1.0051 and itlsec1 = 0.2168. From the positive sign of the estimated i/13ec1 = 0.2168, we conclude the swing caused by initial condition (0 .3,0.0,0.3,0.0) , which is deviated from equilibrium point, is transiently chaotic. It is meaningful to note that if one integrates this system from initial point (0.3 ,O. 0,O. 3,O. 0) over a longer period, then one finds that the angles oscil- late quitely chaotically very long shown in Fig.6 and Fig.7. One finds, surprisingly, the system suddenly loses stability after around 200sec shown in Fig.8. This phenomena demon- strates that transient chaos is not necessarily chaos(since chaos is a bounded behavior) and suggests that transient chaos has a close relationship with long-t,erm instability. In this simple numerical test, the presented method shows that the transient chaos is identified over a short window, 1 sec. If transient*chaos would cause long-term instability, then the estimated XiNaT1 can be used as transient stability index to predict long-term instability over a short time period.

6 Conclusions In this paper, we introduce the concepts of transiently chaotic swing and windowed Lyapunov exponents t,o power system dynamics. More specifically, it is the first pro- posal to use the estimat,ed the largest windowed Lyapunov exponent,itNATl, as on-line detection index. An efficient computation algorithm, which is appropriate for power systems, to estimate X i N A T l is proposed. Two illustrative numerical examples demonstrate the potential of the method to detect transiently chaotic swings and even to predict long- term instability in a very short observation window of real- time measurements. More future work is needed to apply

1.8, 1

1.6

1.4

1.2

0.6

a4

az

Figure 6: Transiently chaotic time-behavior of angle of three-machine system integrated from initial condition (0.3,0.0,0.3,0.0), simulation time=150sec

as

a4

a3

ai

1 .a:

-az

-a3

-a4

-as 0 0.2 0.4 a6 0.8 1 1.2 1.4 1.6 1.8

Figure 7: 2-dimensional projection on the plane (63, us) of transiently chaotic trajectory of three-machine system in tegnted from initial condion (0.3,0.0,0.3,0.0), simulation time=150sec

6

I 4 -

8 :I 0

Figure 8: Sudden loss of bound of transiently chaotic swing integrated from initial condition (0.3,0.0,0.3,0.0), simulation time=250sec

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analysis of complicated behavior of power systems, control of power systems using real-time measurements.

James. S. Thorp (S’58, M’63, SM’80, F’89) received the B.E.E., M.S and Ph.D. degrees from Cornell Univer- sity, Ithaca, NY. In 1962 he joined the faculty at Cornell, where he is currently a Professor of Electrical Engineering. In 1976-1977 he was a Faculty Intern at the American Elec- tric Power Service Corporation, NY, where he is currently a consultant. In 1988 he was an Overseas Fellow at Churchill College, Cambridge, England. He is a member of the IEEE Power System Relaying Committee, Chairman of a Working Group on ”Feasibility of Adaptive Protection and Control” and a member of CIGRE Working Groups on Digital Prc- tection.

the method to real power systems.

References [l] J.S. Thorp, A.G. Phadke, S.H. Horowit,z and M.M. Be-

govic,”Some Applications of Phasor Measurements to Adaptive Protection”, IEEE Trans. on Power Systems, vo1.3, No.2 pp.791-798, May, 1988.

[2] J.S. Thorp,”Control of Electric Power Systems using Real-Time Measurements”, IEEE Control Systems Mag- azine, pp.39-45, Jan., 1989.

[3] Working Group H-7 of the Relaying Channels, Sub- committee of the IEEE Power System Relaying, Work- ing Group Report, ”Synchronized Sampling and Phasor Measurements for Relaying and Control”, to appear in IEEE Tran. on Power Systems.

[4] Bih-Yuan Ku, Jin Lu, Robert J. Thomas and James S. Thorp,” Real-Time prediction of Power System Transient Stability Swings Using On-line Phasor Measurements”, submitted t.0 1993 PICA conference.

[5] Fathi M.A. Salam, Shi Bai, and Shixiong Guo ”Chaotic Dynamics Even in The Highly Damped Swing Equations of Power Systems”, Conference on Decision and Control, pp.681-683, Dec., 1988.

[6] N.Kopel1 and R.B.Washburn,“Chaotic motions in the two degree- of-freedom swing equations”, IEEE Trans. on Circuit and Systems, Vol. CAS-29, pp. 738-746, Nov., 1982.

[7] M. Varghese and J.S. Thorp,”An Analysis of Truncated Fractal Growths in the Stability Boundaries of Three- Node Swing Equations”, IEEE Tram Circuit and Sys- tems, vo1.35, No.7, pp.825-834, July, 1988.

[8] Hsiao-Dong Chiang, Chih-Wen Liu, P.P. Varaiya, Felix F. Wu and Mark G. Lauby,”Chaos in a Simple Power System”, IEEE Winter Power Meet,ing, Feb.1992. Meet- ing).

[9] A.G. Phadke and J.S. Thorp, Computer Relaying for Power Systems, Research Studies Press, 1988.

[lo] T.S. Parker and L.O. Chua, Practical Numerical Al- gorithms for Chaotic Systems, New York, NY:Springer- Verlag, 1989.

[ll] J. Guckenheimer and P.Holmes, Nonlinear Oscillations, Dynamical Systems and Bifirrcations of Vector Fields, New York, NY:Springer-Verlag, 1986, Second printing.

[12] Gene H. Golub and Charles F. Van Loan, hfatrir Corn- putations, second edition, The Johns Hopkins University Press, 1989.

[13] C. Counan, M. Trotignon, E. Corradi, B. Bort,oni, M. Stubbe and J. Deuse, ”Major Incidents on the French Elect,ric System: Pot,ent,ially and Curative Mea.sures Studies” 92 SM 432-5 PWRS.

Chih-Wen Liu received the B.S. degree in Electrical En- gineering from Nat,ional Taiwan University, Taipei, Taiwan, in 1987. He received M.S. degree in Elert,rical Engineering from Cornell University in 1992. Current,ly, he works toward the Ph.D. degree a t Cornell. His research interest,s include

Jin Lu received the B.S. and M.S. degrees in electrical engineering from Nanjing Institute of Technology, Nanjing, P.R.China, in 1982 and 1985, respectively. He received the Ph.D. degree in the Department of Electrical Engineering, Cornell University, Ithaca, NY 14853, in 1990. He is currently an electrical engineer in ORA corporation, a computer software firm. His research interests are in the areas of control and optimization of large scale systems with emphasis on power systems.

Robert J . Thomas (S’66, M’73, SM’82) was born in De- troit, Michigan in 1942. He received the B.S.E.E., M.S.E.E., and Ph.D. degrees from Wayne State University in 1968, 1969, and 1973 respectively. He is currently a Professor of Electrical Engineering a t Cornell University. His research interests include analysis and control of large-scale power systems.

Hsiao-Dong Chjang received the Ph.D. degree in Elec- trical Engineering and computer science from the University of California at Berkeley in 1986, and then worked at the Pacific Gas and Electricity Company on a special project. He joined the Cornell faculty in 1987. He was a recipient of the Engineering Research Initiation Award( 1988) from the National Science Foundation. His research interests include power systems, nonlinear systems, optimization theory and neural networks.

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DISCUSSION

VITALY A. FAYBISOVICH, The R.E.M. E n g i n e e r i n g Co., I n c . , L o s Angeles , CA. :

T h i s and t h e o r e t i c a l q u e s t i o n a b o u t real t i m e i d e n t i f i c a t i o n of t h e t r a n s i e n t l y c h a o t i c swings (TCS) i n power s y s t e m s . I t i n i t i a t e s a t least two q u e s t i o n s :

1. F o r which real c o n d i t i o n s t h e TCS may be o b s e r v e d i n power sys tem?

2 . Can t h e TCS be i d e n t i f i e d by t h e proposed a l g o r i t h m s ?

From o u r p o i n t of v iew t h e TCS is t h e r e s u l t o f n o n l i n e a r o s c i l l a t i o n s i n t h e m u l t i v a r i a b l e low damped s t a b l e sys tem o p e r a t i n g v e r y c l o s e t o t h e s t a b i l i t y boundary . I n t h e a b s e n c e o f random f a c t o r s t h i s s y s t e m ’ s b e h a v i o r i s f u l l y d e t e r m i n i s t i c and c a n be predicted. The c h a o s f e a t u r e s are a r i s i n g due t h e random d i s t u r b a n c e s of dynamic p r o c e s s which i s v e r y s e n s i t i v e t o small v a r i a t i o n s of t h e i n i t i a l c o n d i t i o n s , etc. I n t h e real power sys tem t h o s e random d i s t u r b a n c e s are produced by v a r i a t i o n of l o a d s and g e n e r a t o r s ’ o u t p u t . I n s i m u l a t i o n s t u d y t h e main s o u r c e of s u c h random b e h a v i o r are e r r o r s o f n u m e r i c a l i n t e g r a t i o n . If real power sys tem i s o p e r a t i n g w i t h t h e r e a s o n a b l e r e s e r v e of s t a b i l i t y ( a b o u t 10-15%) t h e TCS i s u n l i k e l y .

I n t h e c lass ical swing e q u a t i o n s t h e a n g l e and f r e q u e n c y o f t h e e l e c t r o m o t i v e f o r c e b e h i n d g e n e r a t o r ’ s t r a n s i e n t r e a c t a n c e are u s e d . From t h e real t i m e p h a s o r measurements t h e a n g l e and f r e q u e n c y a t t h e g e n e r a t o r ’ s b u s are d e f i n e d . Those sets are two d i f f e r e n t sets. For t h i s r e a s o n t h e proposed a l g o r i t h m c a n ‘ t be u s e d for t h e real t i m e TCS i d e n t i f i c a t i o n .

paper c o v e r s v e r y f a s c i n a t i n g p rac t i ca l

W e would appreciate t h e a u t h o r s ‘ comments on t h e above o b s e r v a t i o n s and q u e s t i o n .

Manuscript received August 11, 1993.

C.-W. Liu, J. S. Thorp, J. Lu, R. J. Thomas, and H.-D. Chiang: The authors thank the discussers for their interest on this paper.

Mr. Faybisovich’s Observations and questions are important. We agree that some transiently chaotic swings (TCS) result from multivariable low damped stable system operating very close to the stability boundary, for example, Case 2 in the simulation results of the paper. There are, however, some other cases which can exhibit TCS with high damping like Case 1 in the simulation results. Moreover, it should be noted that TCS like chaos is product of complex intrinsic features of deterministic nonlinear system. Usually the onset of non-empty transversality of stable manifold and unstable manifold of invariant set in system may cause TCS [p. 188,1]. TCS is not product of external disturbances in system like load fluctuations. We also agree the real-time phasor measurements of the angles and frequencies at the generator’s bus are different from the angles and frequencies of the electromotive force behind generator’s transient reactance. But we don’t agree that the proposed algorithm can’t be used for real-time TCS identification. The reason is that the two sets of angles and frequencies mentioned above can be related by static transient reactances and this relationship can be regarded as invertible change of coordinates which will not change the sign of A[NATl used as a detection index of TCS.

Reference

[ 11 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, New York, NY: Springer-Verlag, 1986, second printing.

Manuscript received September 29, 1993.

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